On Thurston’s metric on Teichmüller space: Hyperbolic and Eulcidean settings

I will give an overview of recent results on Thurston’s theory of best Lipshitz Maps between surfaces and his metric on the Teichmüller spaces, adapted to various kinds of surfaces.
In particular I will describe joint works with Ken’ichi Ohshika, Hideki Miyachi and Ismail Saglam, in the case of surfaces equipped with Euclidean metrics. I will mention some open problems.